The_Cambridge_Handbook_of_Physics_Formulas

92 Quantum physics
Dirac notation
∫
a nm = ψ
Matrix element nâψ ∗ a m dx (4.31)
= 〈n|â|m〉 (4.32)
n,m eigenvector indices
a nm matrix element
ψ n basis states
â operator
x spatial coordinate
Bra vector bra state vector = 〈n| (4.33) 〈·| bra
Ket vector ket state vector = |m〉 (4.34) |·〉 ket
∫
Scalar product 〈n|m〉 =
ψ ∗ nψ m dx (4.35)
if Ψ = ∑ c n ψ n (4.36)
n
Expectation
then 〈a〉 = ∑ ∑
c ∗ nc m a nm (4.37)
m n
a The Dirac bracket, 〈n|â|m〉, can also be written 〈ψ n |â|ψ m 〉.
Ψ
c n
wavefunction
probability amplitudes
4.3 Wave mechanics
Potential step a
incident particle
i
V 0
V (x)
ii
Potential
function
Wavenumbers
Amplitude
reflection
coefficient
Amplitude
transmission
coefficient
Probability
currents b
V (x)=
{
0 (x